If the midpoint of the line PQ is (2,3) and the point P is (-2, 1), find the coordinate of the point Q.

Analysis

To find the coordinates of point Q, we use the Midpoint Formula. The midpoint $M(x, y)$ of a line segment connecting two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is calculated as:

Concept

We are given the midpoint $M(2, 3)$ and one endpoint $P(-2, 1)$. We need to find the coordinates of the other endpoint $Q(x_2, y_2)$. We can set up two separate equations—one for the x-coordinate and one for the y-coordinate—and solve for $x_2$ and $y_2$.

Step-by-Step Solution

1. Set up the equation for the x-coordinate:

Multiply both sides by 2:

Add 2 to both sides:

2. Set up the equation for the y-coordinate:

Multiply both sides by 2:

Subtract 1 from both sides:

3. Combine the coordinates: The coordinates of point Q are $(6, 5)$.

Conclusion

Comparing our result $(6, 5)$ with the given options, it matches option D.

Exam Tip

When dealing with midpoint problems, you can use a "jump" method: look at how much the x-coordinate changed from P to the midpoint, and apply that same change again to get to Q.

• From $x = -2$ to $x = 2$ is a jump of $+4$. Adding $+4$ to $2$ gives $6$.
• From $y = 1$ to $y = 3$ is a jump of $+2$. Adding $+2$ to $3$ gives $5$. This is a quick way to verify your answer in the exam hall!